\section{Insurer Operating Results} 
\label{sec:InsurerOperatingResults}

This paper focuses on a single year of insurance operations, considering insurers' revenues and costs, when all insurers and health care providers operate as efficiently as possible with current resources, and all insurers randomly select policyholders from the same population. Efficient health care providers engage in continuous patient monitoring, early diagnosis and treatment, treating patients at minimal cost by using the latest technology, proven treatment protocols, the most appropriate drugs, and appropriate referrals to specialists and alternative resources. Efficient health care does not mean there are no variations in costs. Patients still ignore recommendations, fail to recognize the significance of symptoms, delay seeking care, have multiple co-morbid conditions, and most high cost care is due to unexpected illnesses and accidents. 

\subsection{Insurer Revenues} 
\label{sec:InsurerRevenues} 

Insurers have many revenue streams, including: Premiums for insurance policies; Investment income from bonds and stocks; Interest income from mortgages, and Rental from leased property. However, the core revenues for insurers are the insurers' premium revenues for insurance services provided during the year and it is insurers' premium revenues that are central to this paper.

\subsection{Claims Costs} \label{sec:ClaimsCosts} 

Insurers' claims costs include all costs to settle claims, including: Claim payments to providers and patients; Claims related legal expenses; Claims department salaries, rent, and utilities; and Court fees and Penalties for wrongful claim denial. Claims costs vary, year to year and portfolio to portfolio, because individual policyholder's health care costs vary. This is why we buy insurance. Large insurer's average claims costs rarely vary by more than 5-20\% of long term average claims costs, while individual policyholders' and small insurers' claims costs may vary by as much as 10,000 times average claims costs. 

Acutely ill or traumatically injured patients may generate health care costs of \$500,000 or more while their policy premiums were only \$4,000. These patient's bills are 125 times their insurance premiums, so they have loss ratios, which I will refer to as Population Loss Ratio Estimates ($PLRE$s), of 125.0000. However, most policyholders generate no costs, or very modest costs, and have Population Loss Ratio Estimates of 0.000, 0.0100, or in very few cases, 0.1000. The beauty and utility of insurance is that when insurers combine the premiums and losses for many policyholders, the ``average'' Population Loss Ratio Estimate per policyholder is very close to the Population Loss Ratio, despite the fact that some policyholders generate very large Population Loss Ratio Estimates, and others generate very modest Population Loss Ratio Estimates.


\subsection{Insurer Operating Expense Ratios} \label{sec:InsurerExpenseRatios}

Insurers' Operating Expenses are the costs for all non-claim activities, including: Commissions, rent, utilities, supplies, legal costs, and employee salaries. Expense ratios are averages of operating expenses, to earned premiums (See Equation \ref{eq:ExpenseRatio}) and vary much less, year to year, or portfolio to portfolio, than Population Loss Ratio Estimates. I will assume that all insurers' expense ratios are 15\% of their premium revenues, to focus on the impact of Population Loss Ratio Estimate variability on operating results.
\begin{eqnarray} \label{eq:ExpenseRatio}
 \textrm{Expense Ratio} & = & \frac{ \textrm{Operating Expenses}}{ \textrm{Earned Premiums}} \\
  & = & 0.15
\end{eqnarray}
\subsection{Insurer Population Loss Ratio Estimates} \label{sec:InsurerPopulationLossRatioEstimates}

Insurers' [Policyholders'] Population Loss Ratio Estimates (See Equation~\ref{eq:PopulationLossRatioEstimate}), ($PLRE_{N}$ [$PLRE_{i}$]), are averages of their total claims costs to their total earned premiums:
 \begin{eqnarray} \label{eq:PopulationLossRatioEstimate}
  \textrm{Population Loss Ratio Estimate} & = & \frac{\textrm{Claims Costs}}{\textrm{Earned Premiums}}
\end{eqnarray}
While every insurer and policyholder is going to produce a population loss ratio estimate at the end of each year. However, neither insurers nor policyholders know what their population loss ratio estimates will be at the start of the year. Healthy policyholders may suffer illness or injury, incurring higher than expected claims costs. Very unhealthy policyholders may recover or die, incurring lower than expected costs. Statisticians and actuaries use standard deviations and  standard errors to quantify the uncertainty about individual policyholder's and individual insurer's future policyholder health care claims costs (i.e. the uncertainty in the value of individual policyholder's and individual insurer's population loss ratio estimates).

The entire population of all possible individual policyholder population loss ratio estimates, $PLRE_i$ is the collection: $PLRE_1, PLRE_2, \ldots, PLRE_n$ where $n$ is the size of the population. The mean, or average value, $PLR$, of all the population loss ratio estimates for the Population Loss Ratio ($PLR$), for the collection of all policyholders' population loss ratio estimates is:
\begin{equation}
 PLR = \frac{1}{n}\sum_{i=1}^n (PLRE_i)
\end{equation} 
The standard deviation for an individual randomly selected policyholder's population loss ratio estimate, $PLRE_i$, from the entire population of population loss ratio estimates, $PLRE_1, PLRE_2, \ldots, PLRE_n$, is:
\begin{equation}
 \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (PLRE_i - PLR)^2}
\end{equation} 
Once we know the standard deviation, $\sigma$, for the set of all future individual policyholder population loss ratio estimates, $PLRE_i$, we can calculate standard errors that measure how close individual insurer's population loss ratio estimates, $PLRE_N$, will be to the population loss ratio. We need only assume that each insurer randomly selects $N$ policyholders' population loss ratio estimates, $PLRE_1, PLRE_2, \ldots, PLRE_N$, from the collection of all $n$ possibilities, where $n$ is much larger than $N$. 

The standard error,  $\sigma_{e_{N}}$, for Insurer N, with portfolio size $N$, describes how close to the population loss ratio, Insurer N's population loss ratio estimates will lie over many years. or, if there are many insurers with portfolio size $N$, how close to the population loss ratio the collection of population loss ratio estimates for these similarly sized insurers will lie. Another interpretation is that standard error, $\sigma_{e_{N}}$, describes how far from the population loss ratio Insurer N's population loss ratio estimates will lie if Insurer N could repeatedly draw $N$ policyholders from the population, and calculate population loss ratio estimates for each such portfolio, and plot the resulting values on a graph.

The standard error, $\sigma_{e_{N}}$, for an insurer with $N$ policyholders is a function of the standard deviation, $\sigma$, for an individual policyholder, and the square root of the insurer's portfolio size, the number $N$, of policyholders it insures:
\begin{eqnarray} \label{eq:StandardErrorsFormulaN}
 \sigma_{e_{N}} & = & \sqrt{\frac{  \frac{1}{n} \sum_{i=1}^{n} (PLRE_i -PLR)^2 } {N}} \\
& = & \qquad \frac{\sigma}{\sqrt{N}}
\end{eqnarray} 
The standard deviation, $\sigma$, for an individual, randomly selected policyholder's Population Loss Ratio Estimate ($PLRE_{i}$) is larger than the standard error, $\frac{\sigma}{\sqrt{N}}$, for a randomly selected insurer's Population Loss Ratio Estimate ($PLRE_{N}$) whenever the insurer has more than one policyholder. An insurer's standard error will be much smaller than the standard deviation for an individual policyholder whenever $N$ is large. 

To become perfectly efficient, an insurer would need to issue infinitely many policies. As an insurer increases the number of policies it writes, the expected value of its portfolio estimate of the variance ($\sigma^2$) for an individual policyholder's population loss ratio estimate will always be $\sigma^2$. However, the insurer's standard error, not its standard deviation, changes as it becomes more efficient and it is the standard error that determines the insurer's efficiency. An infinitely large insurer, has standard error 0.0000 = $\frac{\sigma}{\sqrt{\infty}}$. Since there is no variation in a perfectly efficient insurers population loss ratio estimate, there is no difference between one of this insurer's Population Loss Ratio Estimates and any other of this insurer's Population Loss Ratio Estimates. A perfectly efficient insurer's Population Loss Ratio Estimate is always exactly equal to the population loss ratio, $PLR$. 

\subsection{Standard Deviations And Standard Errors} \label{sec:StandardDeviationsandStandardErrors}

That the expected value of the sample standard deviation for an individual, randomly selected policyholder's Population Loss Ratio Estimate is $\sigma$ independent of the size of the sample producing the sample standard deviation causes endless problems for struggling statistics students and health care (finance) policy analysts. Some people have incorrectly assumed that the operating results for large and small health insurers, and large and small insurance risk assuming health care providers, are similar, when this is simply not true. 

Quite the contrary, small insurers produce population loss ratio estimates that tend to lie very far from the population loss ratio while large insurers tend to produce population loss ratio estimates that tend to be very close to the population loss ratio. Professional Caregiver Insurance Risk examines how these differences in insurers' standard errors affects insurers' and how small portfolio size related insurer inefficiencies affect risk assuming health care providers.

Insurers' Population Loss Ratio Estimate standard errors decrease, and the accuracy of their estimates of the population loss ratio increase, as their portfolio sizes increase. Or, phrased differently, Insurers' Population Loss Ratio Estimate standard errors increase, and the accuracy of their estimates of the population loss ratio decrease, as their portfolio sizes decrease. Large insurers' Population Loss Ratio Estimates more accurately estimate Population Loss Ratios than small insurers', solely because their standard errors are smaller. 

We can see how this happens if we consider Equation~\ref{eq:StandardErrorsFormulaN} for two insurers, M and N, insuring $M$ and $N$ policyholders, where $M$ = 100 * $N$. Insurer M's standard error is:
\begin{equation} \label{eq:StandardErrorsFormulaM}
 \sigma_{e_{M}} =  \sqrt{\frac{  \frac{1}{n} \sum_{i=1}^{n} (PLRE_i -PLR)^2 } {M}}
\end{equation} 
and Insurer N's Standard error is as described in Equation~\ref{eq:StandardErrorsFormulaN}. But since $M$ is 100 times as large as $N$, we can rewrite Equation~\ref{eq:StandardErrorsFormulaM} as:
\begin{eqnarray}  \label{eq:StandardErrorsFormulaMeq100NArray}
 \sigma_{e_{M}} & = & \sqrt{\frac{  \frac{1}{n} \sum_{i=1}^{n} (PLRE_i -PLR)^2 } {M}} \\
 & = & \sqrt{\frac{\frac{1}{n} \sum_{i=1}^{n} (PLRE_i -PLR)^2 } {100 * N}} \\
 & = & \frac{1}{10} *  \sigma_{e_{N}}
\end{eqnarray} 
The standard error for Insurer M is $\frac{1}{10}^{\textrm{th}}$ the size of Insurer N's standard error! 

If 95\% of the population loss ratio estimates for Insurer N fall in the interval ($PLR - 1.96 * \sigma_{e_{N}}$, $PLR - 1.96 * \sigma_{e_{N}}$), then about 95\% of the population loss ratio estimates for Insurer M will fall in the interval ($PLR - 1.96 * \frac{\sigma_{e_{N}}}{10}$, $PLR + 1.96 * \frac{\sigma_{e_{N}}}{10}$). 

In particular, if about 95\% of insurers the size of insurer M have loss ratios lying in the interval (0.7000, 0.8000), about 95\% of insurers the size of insurer N will have loss ratios lying in the interval (0.2500, 1.2500). 

This paper compares insurer performance using portfolio size adjusted standard errors (See Section~\ref{sec:QuantitativeAnalysisofInsurerOperations} and Table~\ref{tab:InsurerOperatingResultsByPortfolioSize}).
